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the body has a higher temperature than the gas.

33] IN THE GASEOUS STATE. 365

Thermal transpiration through an aperture in a thin plate.

111. In this case, since there is no tangential stress, we have (Art. 87)

tr=o.

Whence by equation (121)

. a= < 122 >-

a 2

Since p = pwe have, integrating equations (120) and (122), respectively

da VTT 2 ~

ft <>* V o^i

(**' _> / /

,- ' (1226).

aa VTT rr

r J/ r z j~ = ~ /i -"

a 4p

,3 TT

G and ^T are constants, such that -= - is the rate at which heat is

ri

carried across a unit of area, and is the rate at which matter is carried

V**

across.

From equation (122 6) we have

H=-2a?G..

Equation (123) can only be approximately true as a 2 is not constant;

therefore the condition [7=0 is not possible, i.e., it is only approximately

fulfilled, whence it follows that p is only approximately constant. The close-

ness of these approximations will depend on the variation of a 2 , and within

the limits of our approximation we may consider the condition to hold.

From equation (123) we see that the direction of flow of gas is opposite

if

to that of the flow of heat, while since a 2 oc -^ , the rate of flow of gas is

proportional to the flow of heat, to the mass of a molecule, and inversely

proportional to T, the absolute temperature.

By equation (48) we have

s dpa

pu = -j=. ,

VTT dx

or since pz 1 is constant

s da

U = -TT ( 124 )

VTT dx

which it may be noticed is of the same form as results from the equation of

transpiration through a tube when p is constant.

366 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

Thermal Impulsion.

112. In this case there is no motion, therefore

u = 0,

whence from equations (48)

This satisfies equation (120).

Substituting from equation (125) in equations (121) and (122)> and

remembering that p x = - - - , (p^U), Art. 82, we find that these

2 v TT dx

equations lead to the same result if pa? is constant.

Putting >! = ^ we have from equation (122)

z

d d*

whence integrating

da = _ V^# 1 6

dx 9r y r z spi

where H has the same value as in Art. 111.

fj/v* {IT*

Remembering that -=- = 1, also considering s constant, we have,

differentiating,

a = VTT H^ _1_ / J. _!.

x z 9 sp l r y r z \r y r z

rr

'J/2 ""^

Also putting r y = r z , a=a when r=oo, and a = a c when r = c, and

integrating equation (126), we have

^*1L2

9 sp^r'

= $-(*-*), ^ (128).

33] IN THE GASEOUS STATE. 367

In a similar way we obtain from equation (121)

*-_** *|['_I H !V da l

dx~ 7r Pl dx\(r y + rJadx\ .................. (129)

whence integrating

p l TT r v r z ) a.

and from (127)

TT

_

a do?

If r be infinite p x = p = p l and -^ = 0. Therefore C^ = and

^T = ^ 52 a^ (131)

which result may be obtained directly from the value of p x> Art. 82.

From equation (128)

8 s 2 a a'

Trr 2 a

(132).

8cs 2 a,- a"

Trr 3 a.

1 da"

T i,- T i , .. a

Putting ^ = 2 a 2 and neglecting -7-

^ -^

as compared with -

* From an abstract of a paper read before the Koyal Society by Professor Maxwell, in April,

1878 (see Nature, May 9, 1878), I see that Professor Maxwell has obtained an expression for this

inequality of pressure or " stress " arising from the inequality of temperature. The result given

by Professor Maxwell is

dx*

where p. is the coefficient of viscosity, the absolute temperature, and x any one of the three

directions x, y, z. This result, when transformed to the present notation, becomes

And if we put, as in equation (80),

we have

P 1 7T T dX

It is thus seen that the two results are identical in form, but that Professor Maxwell makes

the pressure just three times as great as that given by equation (133).

In the abstract published in Nature, Maxwell has not given the details of the method by which

he arrived at his result.

368 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

/I /^\

s 2 V M V M

8 s 2

TT r 2

(134).

/^- It-

V M V M

8cs*V M V M

TT r

NM

M

From equation (127) we have

p*^P} = & l_Hsr y + r z ..(135)

P! 9 VTT pa r/r/

/?/"

where, as before, ~ - is the quantity of heat carried across a unit of surface.

2r y r z

At points near to the surface.

113. In equations (131), (132), and (135) no account has been taken of

the discontinuity in the immediate neighbourhood of the surface ; hence the

results obtained from these equations may not hold good within the layer of

gas of thickness s, which is adjacent to the surface.

In order to take this discontinuity into account, the equations of steady

conditions should be modified in the manner described in Art. 84, but for

this particular case the same thing may be accomplished in a somewhat

simpler manner.

Suppose the solid surface to be either spherical or cylindrical at the point

ft

considered, and put Cj for the radius. Then it is obvious that when - is

S

very large the pressure on the surface will be but slightly affected by the

layer immediately adjacent to the surface, i.e., putting p Ct for the pressure at

the surface, and p Cl +s f r the pressure at a distance s from the surface,

fv\ _ /y\ s*

I } ILL is small when - is large.

Pc l+ s~Pl S

When, however, the gas surrounding the surface is limited by another

surface, (which for simplicity may be taken concentric and of radius c 2 ),

then in order that 1 ' ^ mav be small, we must have 2 large as

Pc t +s-pc 3 -s s

well as .

s

Our equations, therefore, may be seen to hold good when the radius of

33] IN THE GASEOUS STATE. 369

the solid surface is large compared with s, and the distance between the

opposite surfaces is also large.

On the other hand, in the limit, when either cjs or (c a c 2 )/s are

very small, p c -pi and p Cl -p c will depend entirely on the action of the gas

within the layer of thickness s immediately adjacent to the surface. In

these cases, however, when c a /s or (cj c 2 )/s are small, the action within

this layer may be easily expressed.

114. Let the temperature of the internal surface (sphere or cylinder) be

such that the mean value of a. for the molecules which rebound from this

surface (considered as a group in a uniform gas) is a Ci ; while the temperature

of the external surface is such that the mean value of a. for the molecules

which rebound is a',

The condition that d/s or (c a c 2 )/s are small, necessitates that the

molecules which come up to the inner surface arrive as from a uniform gas

such that a = a'. That is to say, none of the molecules which rebound from

the inner surface can return until their characteristics have been completely

modified by the external surface. For if (cj c 2 )/s is small, the molecules

will cross the interval between the surfaces without encounter, while if GJ/S

is small, although (d c 2 )/s may be large, the characteristics of the gas will

be but slightly affected by the internal layer at a distance s from that

surface, and, by theorem II., the approaching molecules will arrive as from

a uniform gas in the mean condition of the gas at a distance s.

I shall first consider the case in which (d - c 2 )/s is small.

The number of molecules which arrive at the inner surface is proportional

to p'af, and the number which rebound is proportional to p e a c , and since the

numbers must be the same we have

pa.' =

I

The momentum imparted to the surface by the incident molecules is ^

and that imparted by the rebounding molecules is ^ , therefore

( 186 >

Since the molecules which rebound from the internal surface all proceed

to the external surface, and the surfaces are concentric, we have

O. R.

C 2 2 4 v

24

370 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

Therefore

a c/

or

p 2 c 2 2 a'

Equation (138) holds whatever may be the value of Cj/s provided

(c 2 cO/s is small, and it also holds when (c 2 GI)/S is large, provided d/s

is small. When c 2 /s is small and (c 2 c,)/s is large Cj may be neglected in

comparison with c 2 , and we have

Equation (139) is almost identical with what equation (132) becomes as s

approaches in value to r. If s = r, then the only difference in those two

equations is in the coefficient. In comparing these equations, however, it must

be noticed that in (132) a is not the same as a c , for a C) only refers to the one

set of molecules those which are receding from the surface, whereas a refers

to both sets.

At the surface when either cjs or (c 2 c^/'s are small

a c , + a'

a = - L 2 '

Whence making this substitution in equation (132), and putting s =r the

coefficient differs from that in equation (139) by S/TT, which shows the extent

to which discontinuity at the surface affects the result.

General equation oj impulsion.

115. From equations (132) and (139) we may form an equation which

will hold for all values of c/s.

For if the surfaces are spherical

/* : /Z

- p' _ Jl c 2 * - tf , ^ ^ , 8 ** , / Cl Cb\i V V[ ( _ t (14()>

And for cylindrical surfaces

/^- /Z

Pj^y _ ji 0,-c, ^ ^ ^ , 4 - i , /Cl c 2 \) V Jf V (1

M

33] IN THE GASEOUS STATE. 371

/C C \ f C C \

where/; [-, -] and/ 6 (- , -) are respectively unity and zero when either -

\o > / \s s / g

or (c 2 -d)/s are zero, and respectively zero and unity when both d/s and

(c 2 Cj)/s are infinite.

Equations (140) and (141) have been obtained on the assumption that

the solid surfaces are either concentric spheres or concentric cylinders. But

these equations indicate what would be the difference of pressure consequent

on a difference of temperature whatever may be the shape of the surfaces,

and particularly so when d/a and (c 2 - c,)/ are finite, which are the most

important cases.

SECTION XII. APPLICATION TO THE EXPERIMENTS WITH THE FIBRE OF

SILK AND THE RADIOMETER.

116. Comparing the equations (140) and (141) with the equation of

transpiration (101), it appears at once that when H is zero these equations

are identical in form. Hence the curves expressing the relation between the

impulsive forces and the density of the gas under any given conditions, would

be of the same character as those expressing the relation between the

inequalities of pressure and density in the case of thermal transpiration

through a particular porous plate, and it is not necessary for me again to

examine this relation.

Besides which, the experiments on impulsion, elaborate as they have

been, furnish nothing like the definite results which I have obtained in the

experiments on thermal transpiration.

117. The principal results to be deduced from experiments other than

those which are contained in this paper, are :

(1) That the force and motion are proportional to the difference of

temperature, which results are seen to follow directly from equations (124)

and (140).

(2) That with a particular instrument the forces increase with the

rarefaction up to a certain point, after which they fall off; this result also

follows directly from the equation (140).

118. Equations (124) and (140) first revealed to me the fact that the

pressure of gas at which the force would become appreciable must vary

inversely as the size of the surface.

242

372 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

From equation (140) it appears that up to a certain point

and since s oc - and p oc p it appears that

ClV

So that with gas at a given density the smaller the surface the greater

would be the intensity of the impulsive force ; and hence I was led to try the

fibre of silk, with which I obtained evidence of the force at densities of half

an atmosphere ; whereas in the radiometer, with vanes something like 500

times as broad as the fibre of silk, the force does not manifest itself until the

density is very small indeed.

Earlier conclusions.

119. The equations (124) and (140) show that both the forces and the

consequent motion are, cceteris paribus, proportional to the heat communicated

from the surface to the gas; for by equation (128) c a' x H where H is

proportional to the heat communicated from the surface to the gas.

The necessity of such a relation was the subject of my former paper.* I

then obtained the formula

and

To translate this into the symbols of the present paper

f=Pc-p',

d=gp

/3 VTT#

V 2 18 c 2 '

TT

According to my intention e should have been equal , but from the

C

manner in which it was obtained it has the value given above (Appendix,

note 5 (&)). Hence we have

" " ' 1 e& ~ '

lo c a

Proc. Roy. Soc., 1874, p. 407.

33] IN THE GASEOUS STATE. 373

The corresponding equation (Appendix, note 5 (a)) derived from equation

(140) is

18

or when - is small

and when - is large

H

8 1 S #

Q-7= r

y VTT c C 2

It thus appears that the present results entirely confirm the previous

results so far as they went ; and the present investigation is a completion,

not a correction, of the former one.

The present investigation shows that, besides being proportional to the

quantity of heat, the force is proportional to the linear divergence of the

lines along which the heat flows ; and hence, if these lines are parallel, no

matter how great may be the difference of temperature, the gas can exert no

pressure above the normal pressure which it will exert on all surfaces alike.

This is the case, whether the heat is communicated to gas or is spent in

causing evaporation from the surface.

The relation between the difference of pressure and the divergence of the

lines of flow affords a clear explanation of the complex phenomena of the

radiometer ; and as these phenomena have attracted a great deal of interest,

I feel that an explanation of them will not be out of place.

Divergence of the lines of flow and the radiometer.

120. We may readily obtain a graphic representation of the results

expressed by equations (124) and (140).

Let AB, fig. 12, be a plate from which heat is being communicated to

the surrounding gas. Then the lines representing the flow of heat, drawn

according to the law of conduction, are shown in the figure.

374

ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER

[33

(1) The shape of these lines depends on the distribution of temperature

over AB.

Fig. 12 shows what the lines would be if AB were hot on one side and

cold on the other, the gas being at the mean temperature and of unlimited

extent.

(2) The distribution of temperature on an opposite surface, or containing

vessel, will also affect the shape of the lines of flow.

Fig. 13 shows the lines between two parallel plates opposite one another,

the inside face, H, being hotter than the opposite face, C, while the gas and

the outside faces of the plates are at the mean temperature of G and H.

Fig. 13.

(3) The shape of the lines will also depend on the shape of the hot

surface, and the nature of the surface as affecting the rate at which it com-

municates heat to the gas.

33]

IN THE GASEOUS STATE.

375

Fig. 14 shows the direction of the lines for a cup-shaped surface, supposed

to be uniformly at a higher temperature than the gas.

Fig. 14.

In all these figures the lines are supposed to be drawn so that the distance

between any two lines is somewhere between s and 2s, so that the excess of

pressure along the lines of flow depends, cceterisparibus, on the angle between

two consecutive lines. Thus the divergence of the lines indicates the excess

of pressure, the excess being, cceteris paribiis, proportional to the square of

the angle of divergence.

The shapes of the curves of flow are independent of the density of the gas,

but the distance between these lines varies inversely as the density; and

since the angle between the lines at distance s increases with s, we see that

the excess of pressure along the lines of flow increases as the density

diminishes, as long as the mean range of the molecules is not limited by the

size of the containing vessel. When this point is reached, there can be no

further increase in the mean range, and the excess of pressure will pass

through a maximum value, and then fall with the density, until the ratio of

the excess of pressure to the mean pressure becomes constant, which it will

be in the limit.

The distribution of the force of impulsion as indicated by the figures.

121. In fig. 12 the divergence of the lines of flow is much greater towards

the edges of the plates than in the centre ; hence the excess of pressure will

be greater towards the edges. In the same way, on the cold side of the plate,

the convergence of the lines of flow is greatest towards the edges, and here

the pressure will be least.

When the density of the gas is such that the width of the plate is large

compared with s, the divergence of the consecutive heat-lines at the middle

of the plate is small, which shows that there would be but little action on

376 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

this part of the plate. At the edges, however, the divergence is greater, and

there must always be action at the edges ; and the smaller the density of the

gas, or the narrower the plate, the more nearly to the middle of the plate will

the inequality of pressure extend. Thus with a very narrow plate, such as a

spider-line, we may have the inequality of pressure all over the plate, although

in the same gas, with a broad plate, the action might only extend to a distance

from the edge equal to the thickness of the spider-line.

Fig. 13 illustrates the paradox which furnished the clue to this theory.

Towards the middle of the plate the heat-lines are parallel, and consequently

the pressure would be equal and opposite on both plates, being the mean

pressure of the gas ; so that, if the plates were of unlimited extent, there

would be 110 change in the pressure on either plate due to the one being hot

and the other cold.

At the edges, however, the heat-lines diverge from the hot plate ; hence

at this point this plate would be subject to an excess of pressure, which would

tend to force the plate back against the mean pressure of the gas on the

outside. At the edges of the cold plate the heat-lines converge on to the

plate ; hence there will be a deficiency of pressure, and the tendency will be

for the pressure at the back to force the plate forward toward the hot plate.

Thus the action is not to separate the plates, but to force them both to move

in the direction of the hotter plate to cause the hot plate to run away, and

the cold plate to follow it.

Fig. 14 shows the inequality of pressure which may exist over a surface,

itself at uniform temperature, but differing from the temperature of the gas.

On the convex side the lines diverge much more rapidly than on the

concave side, and hence the inequality of pressure due to the communication

of heat will be greater on the convex side.

Stability of the equilibrium.

122. The figures give the lines of flow on the supposition that the gas is

at rest and the surfaces all rigidly fixed. The condition would then be one

of equilibrium. But in order that such a condition might be maintained, it

would be necessary that the condition should be one of stable equilibrium.

This is a point on which the foregoing reasoning furnishes us with no

information.

It is satisfactory, therefore, to be able to see what must happen if the

equilibrium is unstable. This is shown by equation (124), which gives the

motion of the gas, so that there may be no forces.

33] IN THE GASEOUS STATE. 377

If either the surface AB, or the containing vessel, be perfectly free to

move, then no inequality of pressure will be possible, but motion must ensue.

Equation (124) shows the law of such motion.

The. motion.

123. The motion is given by

s da.

*Jirdx

If we suppose the containing vessel to be fixed, then, to allow of the motion

of the gas, the plate must move with the gas. On the other hand, if the

plate be held, the vessel will be carried by the gas in the opposite direction.

Such is the phenomena of the radiometer. The vanes are as nearly as

possible free to move in the vessel, so that their motion merely indicates the

motion of the gas caused by the inequality of temperature in the gas, which

inequality is maintained by the unequal temperature of the two sides of the

vanes arising from their different power of absorbing light, or, in the case of

curved vanes, by the greater temperature of the vanes as compared with the

vessel.

The constraint which is put upon the vanes in a rotatory manner neces-

sarily disturbs somewhat the free motion of the gas, as must also the friction

of the pivot and other resistances. Therefore the condition of the gas within

the vessel cannot be one of absolutely equal pressure ; arid when either the

size of the vanes or the density of the gas are too great, the utmost inequality

of pressure is insufficient to overcome these resistances, and there is no

motion. If, then, exhaustion proceeds, the inequality of pressure increases,

and motion ensues the rate of which, if the vanes were absolutely free,

would increase as the density diminished, until the mean range was limited

by the size of the envelope, so that the larger the envelope the greater the

possible rate of motion. When the paths of the molecules are limited by the

size of the vessel, the motion would, if the vanes were perfectly free to move,

remain constant for all further exhaustion ; but the inequalities of pressure

which the gas is capable of exerting diminish with the further rarefaction,

and hence, in time, must cease to be sufficient to overcome the resistances to

which the motion of the vanes is subject, and then the motion ceases.

124. There are many other points about the phenomena of the radiometer,

but with most of these I have already dealt in my former papers, the reasoning

of which, so far as it goes, appears to me to be perfectly consistent with the

more complete view of the action to which I have now attained.

My chief object in introducing the phenomena of the radiometer in this

378 ON CERTAIN DIMENSIONAL PROPERTIES OB' MATTER [33

paper has been to bring out how completely impulsion belongs to the same

class of actions as thermal transpiration, and the other phenomena depending

on the relation which the size of the external objects bears to the mean range

within the gas.

The action does not depend on the distance between the hot and cold plates.

It has been supposed by some writers on the radiometer, that the action

depends essentially on the distance between the vanes and the sides of the

vessel. This distance, however, is now seen not to be of primary consequence,

as no action will result, however close the plates may be, unless they are of

limited extent of sizes comparable with the mean ranges.

SECTION XIII. SUMMARY AND CONCLUSION.

125. The several steps in this investigation have now been described in

detail. They may be summarized as follows :

(1) The primary step from which all the rest may be said to follow is

the method of obtaining the equations of motion, so as to take into account

not only the normal stresses which result from the. mean motion of the

molecules at a point, but also the normal and tangential stresses which

result from a variation in the condition of the gas (assumed to be molecular).

This method is given in Sections VI., VII., and VIII.

(2) The method of adapting these equations to the case of transpiration

through tubes or porous plates is given in Section IX. The equations of

steady motion being reduced to a general equation, expressing the relation

between the rate of transpiration, the variation of pressure, the variation of

temperature, the condition of the gas, and the dimensions of the tube.

In Section X. is shown the manner in which were revealed the probable

existence (1) of the phenomena of thermal transpiration, and (2) the law of

correspondence between all the results of transpiration with different plates,

so long as the density of the gas is inversely proportional to the lateral linear

dimensions of the passage through the plate ; from which revelations originated

the idea of making experiments on thermal transpiration and transpiration

under pressure.

(3) The method of adapting the equations of steady motion to the case

of impulsion is given in Section XI.

In Section XII. is shown how it first became apparent that the extremely

low pressures at which alone the phenomena of the radiometer had been

obtained were consequent on the comparatively large size of the vanes, and

33] IN THE GASEOUS STATE. 379

that by diminishing the size of the vanes similar results might be obtained

at higher pressures ; whence followed the idea of using the fibre of silk and

33] IN THE GASEOUS STATE. 365

Thermal transpiration through an aperture in a thin plate.

111. In this case, since there is no tangential stress, we have (Art. 87)

tr=o.

Whence by equation (121)

. a= < 122 >-

a 2

Since p = pwe have, integrating equations (120) and (122), respectively

da VTT 2 ~

ft <>* V o^i

(**' _> / /

,- ' (1226).

aa VTT rr

r J/ r z j~ = ~ /i -"

a 4p

,3 TT

G and ^T are constants, such that -= - is the rate at which heat is

ri

carried across a unit of area, and is the rate at which matter is carried

V**

across.

From equation (122 6) we have

H=-2a?G..

Equation (123) can only be approximately true as a 2 is not constant;

therefore the condition [7=0 is not possible, i.e., it is only approximately

fulfilled, whence it follows that p is only approximately constant. The close-

ness of these approximations will depend on the variation of a 2 , and within

the limits of our approximation we may consider the condition to hold.

From equation (123) we see that the direction of flow of gas is opposite

if

to that of the flow of heat, while since a 2 oc -^ , the rate of flow of gas is

proportional to the flow of heat, to the mass of a molecule, and inversely

proportional to T, the absolute temperature.

By equation (48) we have

s dpa

pu = -j=. ,

VTT dx

or since pz 1 is constant

s da

U = -TT ( 124 )

VTT dx

which it may be noticed is of the same form as results from the equation of

transpiration through a tube when p is constant.

366 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

Thermal Impulsion.

112. In this case there is no motion, therefore

u = 0,

whence from equations (48)

This satisfies equation (120).

Substituting from equation (125) in equations (121) and (122)> and

remembering that p x = - - - , (p^U), Art. 82, we find that these

2 v TT dx

equations lead to the same result if pa? is constant.

Putting >! = ^ we have from equation (122)

z

d d*

whence integrating

da = _ V^# 1 6

dx 9r y r z spi

where H has the same value as in Art. 111.

fj/v* {IT*

Remembering that -=- = 1, also considering s constant, we have,

differentiating,

a = VTT H^ _1_ / J. _!.

x z 9 sp l r y r z \r y r z

rr

'J/2 ""^

Also putting r y = r z , a=a when r=oo, and a = a c when r = c, and

integrating equation (126), we have

^*1L2

9 sp^r'

= $-(*-*), ^ (128).

33] IN THE GASEOUS STATE. 367

In a similar way we obtain from equation (121)

*-_** *|['_I H !V da l

dx~ 7r Pl dx\(r y + rJadx\ .................. (129)

whence integrating

p l TT r v r z ) a.

and from (127)

TT

_

a do?

If r be infinite p x = p = p l and -^ = 0. Therefore C^ = and

^T = ^ 52 a^ (131)

which result may be obtained directly from the value of p x> Art. 82.

From equation (128)

8 s 2 a a'

Trr 2 a

(132).

8cs 2 a,- a"

Trr 3 a.

1 da"

T i,- T i , .. a

Putting ^ = 2 a 2 and neglecting -7-

^ -^

as compared with -

* From an abstract of a paper read before the Koyal Society by Professor Maxwell, in April,

1878 (see Nature, May 9, 1878), I see that Professor Maxwell has obtained an expression for this

inequality of pressure or " stress " arising from the inequality of temperature. The result given

by Professor Maxwell is

dx*

where p. is the coefficient of viscosity, the absolute temperature, and x any one of the three

directions x, y, z. This result, when transformed to the present notation, becomes

And if we put, as in equation (80),

we have

P 1 7T T dX

It is thus seen that the two results are identical in form, but that Professor Maxwell makes

the pressure just three times as great as that given by equation (133).

In the abstract published in Nature, Maxwell has not given the details of the method by which

he arrived at his result.

368 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

/I /^\

s 2 V M V M

8 s 2

TT r 2

(134).

/^- It-

V M V M

8cs*V M V M

TT r

NM

M

From equation (127) we have

p*^P} = & l_Hsr y + r z ..(135)

P! 9 VTT pa r/r/

/?/"

where, as before, ~ - is the quantity of heat carried across a unit of surface.

2r y r z

At points near to the surface.

113. In equations (131), (132), and (135) no account has been taken of

the discontinuity in the immediate neighbourhood of the surface ; hence the

results obtained from these equations may not hold good within the layer of

gas of thickness s, which is adjacent to the surface.

In order to take this discontinuity into account, the equations of steady

conditions should be modified in the manner described in Art. 84, but for

this particular case the same thing may be accomplished in a somewhat

simpler manner.

Suppose the solid surface to be either spherical or cylindrical at the point

ft

considered, and put Cj for the radius. Then it is obvious that when - is

S

very large the pressure on the surface will be but slightly affected by the

layer immediately adjacent to the surface, i.e., putting p Ct for the pressure at

the surface, and p Cl +s f r the pressure at a distance s from the surface,

fv\ _ /y\ s*

I } ILL is small when - is large.

Pc l+ s~Pl S

When, however, the gas surrounding the surface is limited by another

surface, (which for simplicity may be taken concentric and of radius c 2 ),

then in order that 1 ' ^ mav be small, we must have 2 large as

Pc t +s-pc 3 -s s

well as .

s

Our equations, therefore, may be seen to hold good when the radius of

33] IN THE GASEOUS STATE. 369

the solid surface is large compared with s, and the distance between the

opposite surfaces is also large.

On the other hand, in the limit, when either cjs or (c a c 2 )/s are

very small, p c -pi and p Cl -p c will depend entirely on the action of the gas

within the layer of thickness s immediately adjacent to the surface. In

these cases, however, when c a /s or (cj c 2 )/s are small, the action within

this layer may be easily expressed.

114. Let the temperature of the internal surface (sphere or cylinder) be

such that the mean value of a. for the molecules which rebound from this

surface (considered as a group in a uniform gas) is a Ci ; while the temperature

of the external surface is such that the mean value of a. for the molecules

which rebound is a',

The condition that d/s or (c a c 2 )/s are small, necessitates that the

molecules which come up to the inner surface arrive as from a uniform gas

such that a = a'. That is to say, none of the molecules which rebound from

the inner surface can return until their characteristics have been completely

modified by the external surface. For if (cj c 2 )/s is small, the molecules

will cross the interval between the surfaces without encounter, while if GJ/S

is small, although (d c 2 )/s may be large, the characteristics of the gas will

be but slightly affected by the internal layer at a distance s from that

surface, and, by theorem II., the approaching molecules will arrive as from

a uniform gas in the mean condition of the gas at a distance s.

I shall first consider the case in which (d - c 2 )/s is small.

The number of molecules which arrive at the inner surface is proportional

to p'af, and the number which rebound is proportional to p e a c , and since the

numbers must be the same we have

pa.' =

I

The momentum imparted to the surface by the incident molecules is ^

and that imparted by the rebounding molecules is ^ , therefore

( 186 >

Since the molecules which rebound from the internal surface all proceed

to the external surface, and the surfaces are concentric, we have

O. R.

C 2 2 4 v

24

370 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

Therefore

a c/

or

p 2 c 2 2 a'

Equation (138) holds whatever may be the value of Cj/s provided

(c 2 cO/s is small, and it also holds when (c 2 GI)/S is large, provided d/s

is small. When c 2 /s is small and (c 2 c,)/s is large Cj may be neglected in

comparison with c 2 , and we have

Equation (139) is almost identical with what equation (132) becomes as s

approaches in value to r. If s = r, then the only difference in those two

equations is in the coefficient. In comparing these equations, however, it must

be noticed that in (132) a is not the same as a c , for a C) only refers to the one

set of molecules those which are receding from the surface, whereas a refers

to both sets.

At the surface when either cjs or (c 2 c^/'s are small

a c , + a'

a = - L 2 '

Whence making this substitution in equation (132), and putting s =r the

coefficient differs from that in equation (139) by S/TT, which shows the extent

to which discontinuity at the surface affects the result.

General equation oj impulsion.

115. From equations (132) and (139) we may form an equation which

will hold for all values of c/s.

For if the surfaces are spherical

/* : /Z

- p' _ Jl c 2 * - tf , ^ ^ , 8 ** , / Cl Cb\i V V[ ( _ t (14()>

And for cylindrical surfaces

/^- /Z

Pj^y _ ji 0,-c, ^ ^ ^ , 4 - i , /Cl c 2 \) V Jf V (1

M

33] IN THE GASEOUS STATE. 371

/C C \ f C C \

where/; [-, -] and/ 6 (- , -) are respectively unity and zero when either -

\o > / \s s / g

or (c 2 -d)/s are zero, and respectively zero and unity when both d/s and

(c 2 Cj)/s are infinite.

Equations (140) and (141) have been obtained on the assumption that

the solid surfaces are either concentric spheres or concentric cylinders. But

these equations indicate what would be the difference of pressure consequent

on a difference of temperature whatever may be the shape of the surfaces,

and particularly so when d/a and (c 2 - c,)/ are finite, which are the most

important cases.

SECTION XII. APPLICATION TO THE EXPERIMENTS WITH THE FIBRE OF

SILK AND THE RADIOMETER.

116. Comparing the equations (140) and (141) with the equation of

transpiration (101), it appears at once that when H is zero these equations

are identical in form. Hence the curves expressing the relation between the

impulsive forces and the density of the gas under any given conditions, would

be of the same character as those expressing the relation between the

inequalities of pressure and density in the case of thermal transpiration

through a particular porous plate, and it is not necessary for me again to

examine this relation.

Besides which, the experiments on impulsion, elaborate as they have

been, furnish nothing like the definite results which I have obtained in the

experiments on thermal transpiration.

117. The principal results to be deduced from experiments other than

those which are contained in this paper, are :

(1) That the force and motion are proportional to the difference of

temperature, which results are seen to follow directly from equations (124)

and (140).

(2) That with a particular instrument the forces increase with the

rarefaction up to a certain point, after which they fall off; this result also

follows directly from the equation (140).

118. Equations (124) and (140) first revealed to me the fact that the

pressure of gas at which the force would become appreciable must vary

inversely as the size of the surface.

242

372 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

From equation (140) it appears that up to a certain point

and since s oc - and p oc p it appears that

ClV

So that with gas at a given density the smaller the surface the greater

would be the intensity of the impulsive force ; and hence I was led to try the

fibre of silk, with which I obtained evidence of the force at densities of half

an atmosphere ; whereas in the radiometer, with vanes something like 500

times as broad as the fibre of silk, the force does not manifest itself until the

density is very small indeed.

Earlier conclusions.

119. The equations (124) and (140) show that both the forces and the

consequent motion are, cceteris paribus, proportional to the heat communicated

from the surface to the gas; for by equation (128) c a' x H where H is

proportional to the heat communicated from the surface to the gas.

The necessity of such a relation was the subject of my former paper.* I

then obtained the formula

and

To translate this into the symbols of the present paper

f=Pc-p',

d=gp

/3 VTT#

V 2 18 c 2 '

TT

According to my intention e should have been equal , but from the

C

manner in which it was obtained it has the value given above (Appendix,

note 5 (&)). Hence we have

" " ' 1 e& ~ '

lo c a

Proc. Roy. Soc., 1874, p. 407.

33] IN THE GASEOUS STATE. 373

The corresponding equation (Appendix, note 5 (a)) derived from equation

(140) is

18

or when - is small

and when - is large

H

8 1 S #

Q-7= r

y VTT c C 2

It thus appears that the present results entirely confirm the previous

results so far as they went ; and the present investigation is a completion,

not a correction, of the former one.

The present investigation shows that, besides being proportional to the

quantity of heat, the force is proportional to the linear divergence of the

lines along which the heat flows ; and hence, if these lines are parallel, no

matter how great may be the difference of temperature, the gas can exert no

pressure above the normal pressure which it will exert on all surfaces alike.

This is the case, whether the heat is communicated to gas or is spent in

causing evaporation from the surface.

The relation between the difference of pressure and the divergence of the

lines of flow affords a clear explanation of the complex phenomena of the

radiometer ; and as these phenomena have attracted a great deal of interest,

I feel that an explanation of them will not be out of place.

Divergence of the lines of flow and the radiometer.

120. We may readily obtain a graphic representation of the results

expressed by equations (124) and (140).

Let AB, fig. 12, be a plate from which heat is being communicated to

the surrounding gas. Then the lines representing the flow of heat, drawn

according to the law of conduction, are shown in the figure.

374

ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER

[33

(1) The shape of these lines depends on the distribution of temperature

over AB.

Fig. 12 shows what the lines would be if AB were hot on one side and

cold on the other, the gas being at the mean temperature and of unlimited

extent.

(2) The distribution of temperature on an opposite surface, or containing

vessel, will also affect the shape of the lines of flow.

Fig. 13 shows the lines between two parallel plates opposite one another,

the inside face, H, being hotter than the opposite face, C, while the gas and

the outside faces of the plates are at the mean temperature of G and H.

Fig. 13.

(3) The shape of the lines will also depend on the shape of the hot

surface, and the nature of the surface as affecting the rate at which it com-

municates heat to the gas.

33]

IN THE GASEOUS STATE.

375

Fig. 14 shows the direction of the lines for a cup-shaped surface, supposed

to be uniformly at a higher temperature than the gas.

Fig. 14.

In all these figures the lines are supposed to be drawn so that the distance

between any two lines is somewhere between s and 2s, so that the excess of

pressure along the lines of flow depends, cceterisparibus, on the angle between

two consecutive lines. Thus the divergence of the lines indicates the excess

of pressure, the excess being, cceteris paribiis, proportional to the square of

the angle of divergence.

The shapes of the curves of flow are independent of the density of the gas,

but the distance between these lines varies inversely as the density; and

since the angle between the lines at distance s increases with s, we see that

the excess of pressure along the lines of flow increases as the density

diminishes, as long as the mean range of the molecules is not limited by the

size of the containing vessel. When this point is reached, there can be no

further increase in the mean range, and the excess of pressure will pass

through a maximum value, and then fall with the density, until the ratio of

the excess of pressure to the mean pressure becomes constant, which it will

be in the limit.

The distribution of the force of impulsion as indicated by the figures.

121. In fig. 12 the divergence of the lines of flow is much greater towards

the edges of the plates than in the centre ; hence the excess of pressure will

be greater towards the edges. In the same way, on the cold side of the plate,

the convergence of the lines of flow is greatest towards the edges, and here

the pressure will be least.

When the density of the gas is such that the width of the plate is large

compared with s, the divergence of the consecutive heat-lines at the middle

of the plate is small, which shows that there would be but little action on

376 ON CERTAIN DIMENSIONAL PROPERTIES OF MATTER [33

this part of the plate. At the edges, however, the divergence is greater, and

there must always be action at the edges ; and the smaller the density of the

gas, or the narrower the plate, the more nearly to the middle of the plate will

the inequality of pressure extend. Thus with a very narrow plate, such as a

spider-line, we may have the inequality of pressure all over the plate, although

in the same gas, with a broad plate, the action might only extend to a distance

from the edge equal to the thickness of the spider-line.

Fig. 13 illustrates the paradox which furnished the clue to this theory.

Towards the middle of the plate the heat-lines are parallel, and consequently

the pressure would be equal and opposite on both plates, being the mean

pressure of the gas ; so that, if the plates were of unlimited extent, there

would be 110 change in the pressure on either plate due to the one being hot

and the other cold.

At the edges, however, the heat-lines diverge from the hot plate ; hence

at this point this plate would be subject to an excess of pressure, which would

tend to force the plate back against the mean pressure of the gas on the

outside. At the edges of the cold plate the heat-lines converge on to the

plate ; hence there will be a deficiency of pressure, and the tendency will be

for the pressure at the back to force the plate forward toward the hot plate.

Thus the action is not to separate the plates, but to force them both to move

in the direction of the hotter plate to cause the hot plate to run away, and

the cold plate to follow it.

Fig. 14 shows the inequality of pressure which may exist over a surface,

itself at uniform temperature, but differing from the temperature of the gas.

On the convex side the lines diverge much more rapidly than on the

concave side, and hence the inequality of pressure due to the communication

of heat will be greater on the convex side.

Stability of the equilibrium.

122. The figures give the lines of flow on the supposition that the gas is

at rest and the surfaces all rigidly fixed. The condition would then be one

of equilibrium. But in order that such a condition might be maintained, it

would be necessary that the condition should be one of stable equilibrium.

This is a point on which the foregoing reasoning furnishes us with no

information.

It is satisfactory, therefore, to be able to see what must happen if the

equilibrium is unstable. This is shown by equation (124), which gives the

motion of the gas, so that there may be no forces.

33] IN THE GASEOUS STATE. 377

If either the surface AB, or the containing vessel, be perfectly free to

move, then no inequality of pressure will be possible, but motion must ensue.

Equation (124) shows the law of such motion.

The. motion.

123. The motion is given by

s da.

*Jirdx

If we suppose the containing vessel to be fixed, then, to allow of the motion

of the gas, the plate must move with the gas. On the other hand, if the

plate be held, the vessel will be carried by the gas in the opposite direction.

Such is the phenomena of the radiometer. The vanes are as nearly as

possible free to move in the vessel, so that their motion merely indicates the

motion of the gas caused by the inequality of temperature in the gas, which

inequality is maintained by the unequal temperature of the two sides of the

vanes arising from their different power of absorbing light, or, in the case of

curved vanes, by the greater temperature of the vanes as compared with the

vessel.

The constraint which is put upon the vanes in a rotatory manner neces-

sarily disturbs somewhat the free motion of the gas, as must also the friction

of the pivot and other resistances. Therefore the condition of the gas within

the vessel cannot be one of absolutely equal pressure ; arid when either the

size of the vanes or the density of the gas are too great, the utmost inequality

of pressure is insufficient to overcome these resistances, and there is no

motion. If, then, exhaustion proceeds, the inequality of pressure increases,

and motion ensues the rate of which, if the vanes were absolutely free,

would increase as the density diminished, until the mean range was limited

by the size of the envelope, so that the larger the envelope the greater the

possible rate of motion. When the paths of the molecules are limited by the

size of the vessel, the motion would, if the vanes were perfectly free to move,

remain constant for all further exhaustion ; but the inequalities of pressure

which the gas is capable of exerting diminish with the further rarefaction,

and hence, in time, must cease to be sufficient to overcome the resistances to

which the motion of the vanes is subject, and then the motion ceases.

124. There are many other points about the phenomena of the radiometer,

but with most of these I have already dealt in my former papers, the reasoning

of which, so far as it goes, appears to me to be perfectly consistent with the

more complete view of the action to which I have now attained.

My chief object in introducing the phenomena of the radiometer in this

378 ON CERTAIN DIMENSIONAL PROPERTIES OB' MATTER [33

paper has been to bring out how completely impulsion belongs to the same

class of actions as thermal transpiration, and the other phenomena depending

on the relation which the size of the external objects bears to the mean range

within the gas.

The action does not depend on the distance between the hot and cold plates.

It has been supposed by some writers on the radiometer, that the action

depends essentially on the distance between the vanes and the sides of the

vessel. This distance, however, is now seen not to be of primary consequence,

as no action will result, however close the plates may be, unless they are of

limited extent of sizes comparable with the mean ranges.

SECTION XIII. SUMMARY AND CONCLUSION.

125. The several steps in this investigation have now been described in

detail. They may be summarized as follows :

(1) The primary step from which all the rest may be said to follow is

the method of obtaining the equations of motion, so as to take into account

not only the normal stresses which result from the. mean motion of the

molecules at a point, but also the normal and tangential stresses which

result from a variation in the condition of the gas (assumed to be molecular).

This method is given in Sections VI., VII., and VIII.

(2) The method of adapting these equations to the case of transpiration

through tubes or porous plates is given in Section IX. The equations of

steady motion being reduced to a general equation, expressing the relation

between the rate of transpiration, the variation of pressure, the variation of

temperature, the condition of the gas, and the dimensions of the tube.

In Section X. is shown the manner in which were revealed the probable

existence (1) of the phenomena of thermal transpiration, and (2) the law of

correspondence between all the results of transpiration with different plates,

so long as the density of the gas is inversely proportional to the lateral linear

dimensions of the passage through the plate ; from which revelations originated

the idea of making experiments on thermal transpiration and transpiration

under pressure.

(3) The method of adapting the equations of steady motion to the case

of impulsion is given in Section XI.

In Section XII. is shown how it first became apparent that the extremely

low pressures at which alone the phenomena of the radiometer had been

obtained were consequent on the comparatively large size of the vanes, and

33] IN THE GASEOUS STATE. 379

that by diminishing the size of the vanes similar results might be obtained

at higher pressures ; whence followed the idea of using the fibre of silk and

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